**Exponential Growth and Decay :**

Exponential growth / decay is a specific way that a quantity may increase / decrease over time.

To solve problems on exponential growth and decay, we have to be aware of exponential growth and decay functions.

Let us consider the following two examples.

When we invest some money in a bank, it grows year by year, because of the interest paid by the bank.

We buy a car and use it for some years. When it becomes too old, we would like to sell it.

In the first example, we will be keen to know the final value (Amount invested + Interest) of our deposit. To know the final value of the deposit, we have to use growth function.

In the second example, we will be eager to know the sale value of the car (Purchased price - depreciation). Here we have to use decay function.

In this way, growth and decay functions are being used in our life.

Many real world phenomena are being modeled by functions which describe how things grow or decay as time passes.

Let us see the formulas which use to estimate and growth and decay.

**Formula 1 :**

The formula given below is related to compound interest formula and represents the case where interest is being compounded continuously.

That is, at any instant the balance is changing at a rate that equals "r" times the current balance.

We use this formula, when it is given "exponential growth/or decay".

**A = Pe ^{rt}**

A ---> Ending amount

P ---> Beginning amount

r ---> Growth/Decay rate

t ---> Time

**Note :**

If it is decay function, the value of "r" will be negative.

**Formula 2 :**

The formula given below is compound interest formula and represents the case where interest is being compounded annually or the growth is being compounded once the term is completed.

**A = P(1 + r) ^{n}**

A ---> Ending amount

P ---> Beginning amount

r ---> Growth/Decay rate

n ---> No. of years / Time

**Note :**

If it is decay function, the value of "r" will be negative.

**Formula 3 :**

The formula given below is related to geometric progression. Here, the initial amount will grow/decay at the constant ratio "b".

**A = ab ^{x}**

A ---> Ending amount

a ---> Beginning amount

b ---> Growth/Decay ratio

x ---> No. of years / terms

**Note :**

If it is growth function, we will have "r" > 1

If it is decay function, we will have 0 < r < 1

**Problem 1 :**

Mark invests $1,500 at a rate of 6% interest compounded annually. How much is the investment worth after 5 years ?

**Solution :**

Compound Interest Formula :

**A =** **P(1 + r) ^{n}**

Substitute

P = 1500

r = 6% or 0.06

n = 5

Then,

A = 1500(1 + 0.06)^{5}

A = 1500(1.06)^{5}

Use a calculator.

A = 2,007.34

So, the worth of the investment after 5 years is about $2,007.34

**Problem 2 :**

The price of a new automobile is $28,000. If the value of the automobile decreases 12% per year, what will be the price of the automobile after 5 years?

**Solution :**

Exponential Decay Formula :

**A =** **P(1 - r) ^{n}**

Substitute

P = 28000

r = 12% or 0.12

n = 5

Then,

A = 2800(1 - 0.12)^{5}

A = 1500(0.88)^{5}

Use a calculator.

A = 14,776.49

So, the price of the automobile will be about $14,776.49

**Problem 3 :**

An investment worth $2500 made in a bank which pays 10% interest per year compounded continuously. What will be the value of the investment after 10 years ?

**Solution :**

We have to use the formula given below to know the value of the investment after 3 years.

**A = Pe ^{rt}**

Substitute

P = 2500

r = 10% or 0.1

t = 10

e = 2.71828

Then,

A = 2500(2.71828)^{(0.1)10}

A = 6795.70

So, the value of the investment after 10 years is $6795.70.

**Problem 4 :**

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture initially, how many bacteria will be present at the end of 8th hour?

**Solution :**

Note that the number of bacteria present in the culture doubles at the end of successive hours.

Since it grows at the constant ratio "2", the growth is based is on geometric progression.

We have to use the formula given below to find the no. of bacteria present at the end of 8th hour.

**A = ab ^{x}**

Substitute

a = 30

b = 2

x = 8

Then, we have

A = 30(2^{8})

A = 30(256)

A = 7680

So, the number of bacteria at the end of 8th hour is 7680.

**Related Topics : **

Apart from the stuff given in this section, if you need any other stuff in math, please use or google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**